I will present joint work with M. Mustata, in which we study a sequence of ideals arising naturally from M. Saito's Hodge filtration on the localization along a hypersurface. The multiplier ideal of the hypersurface appears as the first step in this sequence, which as a whole provides a more refined measure of singularities. We give applications to the comparison between the Hodge filtration and the pole order filtration, adjunction, and the singularities of hypersurfaces in projective space and theta divisors on abelian varieties.